complex analysis paper–ii segment wise questions from 1983 to 2011 civil service

Previous Years Questions (19832011) Segment-wise Complex Analysis Paper II

1983

Obtain the Taylor and Laurent series expansions which represent the function 2 1

2 3z

z z in the regions (i)|z|

1988

By evaluating 2

dzz

over a suitable contour C, Prove that 0

1 2 cos5 4 cos

d (1997)

If f is analytic in |Z| ( )z R

f z dzz x z y

and deduce that a function

analytic and bounded for all finite z is a constant.

If 0

nn

n

f z a z has radius of convergence R and prove that

Evaluate , if a lies inside the closed contour C.

Prove that by the integrating along the boundary of the rectangle |x|

(1997)

Prove that the coefficients of the expansion satisfy Determine .nC

1989

Find the singularities of 1sin1 z

in the complex plane.

1990

Let f be regular for |Z| < R , prove that , if 0

If show that 1 12

zze 20 1 2J t zJ t z J t 1 2 32 3

1 1 1J t J t J tz z z

Examine the nature of the singularity of at infinity Evaluate the residues of the function at all singularities and show that their sum is zero.

By integrating along a suitable contour show that 1 sin

ax

x

ee a

where 0

Find the residues of at all its poles in the finite plane.

By means of contour integration, evaluate 2

20

(log )1

e u duu

.

1995 Let . Prove that u is a harmonic function. Find a harmonic function v such that u+iv is an

analytic function of z.

Find the Taylor series expansion of the function 4 9zf z

z around Z=0. Find also the radius of convergence of the obtained series.

Let C be the circle | Z |=2 described contour clockwise .Evaluate the integral 2

cosh1c

z dzz z

Let a 20

cos1

axdxx

with the aid of residues. (2006)

Let f be analytic in the entire complex plane. Suppose that there exists a constant A > 0 such that |f(z)| Prove that there exists a complex number a such that f(z)=az for all Z.

Suppose a power series converges at a point . Let 1z be such that show that the series

converges uniformly in the disc 1: .z z z

1996

Evaluate 2

0

1 cossinz

zzLt

Show that Z=0 is not a branch point for the function sin( ) .zf zz

Is it a removable singularity ?

Prove that every polynomial equation 20 1 2 ........ 0, 0 1n

n na a Z a Z a Z a n has exactly n roots.

By using residue theorem, evaluate 2

20

log 11

e x dxx

About the singularity Z=-2, find the Laurent expansion of 13 sin2

zz

. Specify the region of convergence and nature of

singularity at Z= -2.

1997

If 1 2 2 n nAA Af z

z a z a z a find the residue at a for

f zz b

where 1 2, ,........... ,nA A A a & b are constant.

What is the residue at infinity.

Find the Laurent series for the function 1

ze in 0 z . Deduce that cos0

1 1cos sin!

e n dn (2001)

Find the function f(z) analytic with in the unit circle which takes the values 2cos sin ,0 2

2 cos 1a ia a on the circle.

Show that the function 3 3

2 2

1 1( ) , 0

x i y if z z

x y

2( ) coszf z e ec z

2 2 3 2( , ) 3 2 2u x y x y x y y

0

nn

na z 0 0Z 1 0 1 0Z Z and Z

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0 0f is continuous and C-R conditions are satisfied at z=0, but ( )f z does not exist at z=0.

Find the Laurent expansion of1 2

zz z

about the singularity Z= -2. Specify the region of convergence and the nature of

singularity at Z=-2

By using the integral representation of (0)nf , prove that 2

1

1 ,! 2 !

n n xz

nc

x x e dzn i n z

where c is any closed contour

surrounding the origin. Hence show that 2 2

2 cos

0 0

1 .! 2

nx

n

x e dn

Using residue theorem 2

0

.3 2cos sin

d

1999

Examine the nature of the function 2 5

4 10 , 0x y x iy

f z zx y

0 0f is a region including the origin and hence show that Cauchy Riemann equations are satisfied at the origin but f(z) is not analytic there.

For the function 21( ) ,

3 2f z

z z find Laurent series for the domain (i) 1 < |z|< 2 (ii) | z | >2 show further that

( ) 0c

f z dz where c is any closed contour enclosing the points Z=1 and Z=2 .

Using residue theorem show that 4sin sin ; 0

4 2ax axdx e a a

x (1984,1998)

The function f(z) has a double pole at z=0 with residue 2, a simple pole at z=1 with residue 2, is analytic at all other finite points of the plane and is bounded as z . If f(2)=5 and f(-1)=2 , find f(z).

What kind of singularities the following functions have? (i) 1 21 z

at z ie

(ii) 1sin cos 4

at zz z

(iii)

2cot z at z a and zz a

. In case (iii) above what happens when a is an integer.(including a=0)?

2000

Suppose ( )f is continuous on a circle C. show that C

fd

z as z varies inside C, is differentiable under the integral

sign. Find the derivative hence or otherwise derive an integral representation for ( )f z if f z is analytic on and inside of C. (30)

2001

Prove that the Riemann Zeta function defined by 1

z

n

z n converges for Re 1z and converges uniformly for

Re 1z where 0 is arbitrary small. (12)

Show that 41

.1 2

dxx



2002

Suppose that f and g are two analytic functions on the set of all complex numbers with 1 1f gn n

for n=1,2,3,..

then show that f(z)=g(z) for each Z in . (12)

Show that when 0 < | z-1 | < 2, the function 1 3

zf zz z

has the Laurent series expansion in powers of z-1 as

20

11 3 .2 1 2

n

nn

zz (15)

2003

Use the method of contour integration to prove that 2 2 20

; 0 .sin 1

ad aa a

(15)

2004 If all zeros of a polynomial p(z) lie in a half plane then show that zeros of the derivative ( )p Z also lie in the same half plane . (15)

Using contour integration , evaluate 2 2

20

cos 3 ,0 1.1 2 cos 2

d pp p (15)

2005 If f(z)=u+iv is an analytic function of the complex variable z and (cos sin )xu v e y y determine f(z) in terms of z.(12)

Expand 11 3

f zz z

in Laurents series which is valid for (i) 1< |z| 3 (iii) |z| < 1. (30)

2006

With the aid of residues, evaluate 20

cos 2 ; 1 1.1 2 cos

d aa a (15)

2007

Prove that the function f defined by 5 , 0

| | 4( )0, 0

z zzf z

z is not differentiable at z = 0 (12)

Evaluate (by using residue theorem) 2

20

.1 8cos

d (15)

2008

Find the residue of 3cot cotz hz

z at z=0. (12)

Evaluate 2

22 2

1log 62 2 4

z

C

e z dzz z z z

where c is the circle |Z|=3. State the theorem you use in evaluating

above integral. (15)



Let f(z) be entire function satisfying 2( ) | |f z k z for some +ve constant K and all Z. show that 2( )f z az for some constant a. (15)

2009

Let 1

0 1 1

0 1

.......( ) , 0.

.......

nn

nnn

a a z a zf z b

b b z b z,

Assume that the zeroes of the denominator are simple. Show that the sum of the residues of f(z) at its poles is equal to 1nn

ab

.

(12) 2 2 2 Show that:

2

2 2 20

2cos sin

d

(30)

2010 Show that 3 2( , ) 2 3u x y x x xy is a harmonic function. Find a harmonic conjugate of u(x, y). Hence find the analytic

function f for which u(x, y) is the real part. (12) (i)Evaluate the line integral .

c

f z dz where 2( ) ,f z z c is the boundary of the triangle with vertices A (0, 0), B (1, 0), C

(1, 2) in that order. (ii). Find the image of the finite vertical strip R : x = 5 to x =9, - plane under the exponential function.(15)

Find the Laurent series of the function 1( ) exp

2n

nn

f z z as c zz

0for z

Where 1 cos sin , 0, 1, 2,nC n d n

( )iz e as contour in this region. (15)

2011

Evaluate by Contour integration,1

1 32 30

dxx x

. (15)

Find the Laurent Series for the function

2

11

f zz

with centre z=1. (15)

Show that the series for which the sum of first n terms 2 2 ,0 11nnxf x xn x

cannot be differentiated term-by-term at x=0.What happens at 0?x (15)

If f(z)=u+iv is an analytic function of z=x+iy and cos sincos cos

ye x xu vhy x

,find f(z) subject to the condition,

3 .2 2

if (12)



complex analysis paper–ii segment wise questions from 1983 to 2011 civil service

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